Well, technically your TA is correct in that there is no cruise ship terminal parking and all of the pay lots around the port are considered "long term airport parking". Being that the port and the airport are so close together, they share the lots, but they are airport lots and typically shuttles run to and from the airport...but some also drop off at the port. Hotels also have park and cruise deals. Our on site airport parking is around $32 per day, so most people use the offsite lots that charge closer to $15 per day. Check Out Park N Fly San Diego. Lot 3 is their cruise ship terminal parking lot and they have a shuttle that will take you to the pier. There used to be parking near the port, but they have since developed that land with the new Waterfront park.
With regards to walking to the port from the airport, i would not do that...the airport is fenced quite a bit for security and then for the construction of the new parking structures, so to walk, you couldn't take the most direct route (that shows up on google earth) because of the fences and lack of sidewalks...and when I mean lack of sidewalks, I mean walking in traffic.